Problem 57 - Algebra and Trigonometry Real Mathematics, Real People

Question

Use a graphing utility to approximate any solutions of the equation. [Remember to write the equation in the form \(f(x)=0.1\) $$\sqrt{x-2}=3$$

Step-by-Step Solution

Answer

After following the above steps, use a graphing utility to plot the equation \(f(x)=(\sqrt{x-2}-3)^2\). The approximate solution is where the graph intersects the x-axis.

Step 1: Rewrite the equation in the form \(f(x)=0\)

Start by rearranging the equation as \(f(x)=\sqrt{x-2}-3\). Now, the equation matches the required format \(f(x)=0\).

Step 2: Simplify the equation

Next, square both sides of the equation to eliminate the square root. This leaves the equation \(f(x)=(\sqrt{x-2}-3)^2\). This equation represents either a parabola or a circle and it is clearer to plot on a graphing utility.

Step 3: Use a graphing utility

Plot the function \(f(x)=(\sqrt{x-2}-3)^2\) on the graphing utility. The x-coordinate at which the graph intersects the x-axis are the roots of the original equation.

Key Concepts

Solving Equations

When confronted with any equation, our overall goal is to find the value(s) of the variable that make the equation true. This is known as solving the equation.
Typically, this involves a series of algebraic manipulations that simplify or transform the equation. If you're asked to solve something like \(\sqrt{x-2} = 3\), you'll start by manipulating the equation to isolate the variable of interest, which in this case is \(x\).
First, rewriting the equation into the form \(f(x)=0\) helps us see it more clearly against a standard for solving equations. For example, by shifting the terms we arrive at \(\sqrt{x-2} - 3 = 0\). By setting the process as a function of \(f(x) = \sqrt{x-2} - 3\), it becomes intuitive to search for values of \(x\) where this function equals zero.
By graphing this function and identifying where it crosses the x-axis, we visually identify the solution(s) of the equation: these are the points at which \(f(x) = 0\). It's a practical method that merges algebra with visual insight, enabling an intuitive understanding of solutions.

Square Roots

Square roots are fundamental in mathematics, representing a value that, when multiplied by itself, yields another number. In this context, \(\sqrt{x-2}\) signifies a number that when squared will give \(x-2\).
To solve equations incorporating square roots like \(\sqrt{x-2} = 3\), we focus on getting rid of the square roots to simplify the solving process. This can be done by squaring both sides of the equation, effectively transforming it into a more straightforward form.
Here, squaring \(\sqrt{x-2} = 3\) yields \((x-2) = 3^2\), which simplifies to \(x - 2 = 9\). Solving this for \(x\) leads us to \(x = 11\).
However, every time you square both sides, it's crucial to verify the results back in the original equation to ensure no extraneous solutions are included. This process underscores the importance of understanding square roots in algebraic functions.

Algebraic Functions

Algebraic functions are a significant component of algebra, describing relationships between variables. These often involve operations like addition, subtraction, multiplication, division, and roots.
In the equation \(\sqrt{x-2} - 3 = 0\), the function \(f(x) = \sqrt{x-2} - 3\) is an example, blending both a root function and linear elements.
By turning this equation into \(f(x)=0\), we convert it into a standard form that can be easily analyzed with graphing utilities. Algebraic functions often represent curves on a graph. They provide a visual tool for understanding how variables are related and for finding solutions.
Using a graphing utility, we gain a valuable perspective by plotting this function to find where it meets the x-axis. This intersection point directly corresponds to the solution of the equation, demonstrating how algebraic functions describe and solve mathematical relationships practically.